I am trying to find a matrix representation in Mat$_{2×2}(\mathbb C)$ for a reflection about a line $z=z(t) = a+bt$ where only $t$ is restricted to be in $\mathbb R$ as a parameter.
I am thinking that a general reflextion can be viewed as a rotation followed by a reflection about the real line followed by inverse rotation, hence it suffices to find a Möbius representation for the reflection about the real line. However, I am not exactly sure how to find this.
If you try to find a Möbius transformation $T(z)=\frac{az+b}{cz+d}$ that fixes the points on a line, you will get the identity, $T(z)=z$. This follows from the fact that a Möbius transformation is determined by what it does to $3$ points. (In fact, a nonidentity Möbius transformation has only two fixed points on the Riemann sphere.)
So reflection in a line can't be expressed as a Möbius transformation.